Separation between the plates of a parallel plate capacitor is `d` and the area of each plates is `A`. When a slab of material of dielectric constant `k` and thickness `t(t lt d)` is introduced between the plates. Its capacitance becomes

Options

A. `(epsilon_(0)A)/(d + t(1-(1)/(k)))`
B. `(epsilon_(0)A)/(d + t(1+(1)/(k)))`
C. `(epsilon_(0)A)/(d - t(1-(1)/(k)))`
D. `(epsilon_(0)A)/(d - t(1+(1)/(k)))`

Explanation

To find the capacitance of a parallel plate capacitor when a dielectric slab is introduced, we can follow these steps:

### Step 1: Understand the Configuration
We have a parallel plate capacitor with:
- Plate separation = \( d \)
- Area of each plate = \( A \)
- Dielectric slab with thickness \( t \) (where \( t < d \)) and dielectric constant \( k \)

### Step 2: Analyze the Voltage Distribution
When the dielectric slab is inserted, the capacitor can be thought of as two capacitors in series:
1. The part with the dielectric slab of thickness \( t \)
2. The part with air (or vacuum) of thickness \( d - t \)

### Step 3: Calculate the Voltage Across Each Section
- **Voltage across the dielectric slab**:
\[ V_{\text{medium}} = \frac{Q}{A \epsilon_0 k} \cdot t \]
- **Voltage across the air gap**:
\[ V_{\text{air}} = \frac{Q}{A \epsilon_0} \cdot (d - t) \]

### Step 4: Total Voltage Across the Capacitor
The total voltage across the capacitor is the sum of the voltages across both sections:
\[ V = V_{\text{air}} + V_{\text{medium}} = \frac{Q}{A \epsilon_0} (d - t) + \frac{Q}{A \epsilon_0 k} t \]

### Step 5: Simplify the Total Voltage Expression
Factoring out \( \frac{Q}{A \epsilon_0} \):
\[ V = \frac{Q}{A \epsilon_0} \left( (d - t) + \frac{t}{k} \right) \]

### Step 6: Calculate the Capacitance
The capacitance \( C \) is defined as:
\[ C = \frac{Q}{V} \]
Substituting the expression for \( V \):
\[ C = \frac{Q}{\frac{Q}{A \epsilon_0} \left( (d - t) + \frac{t}{k} \right)} \]
This simplifies to:
\[ C = A \epsilon_0 \left( \frac{1}{(d - t) + \frac{t}{k}} \right) \]

### Step 7: Final Expression for Capacitance
Thus, the capacitance of the parallel plate capacitor with the dielectric slab is:
\[ C = \frac{A \epsilon_0}{(d - t) + \frac{t}{k}} \]